Unselectable edge? Help!
It looks as though you have a non-planar face.  Since there is no such thing as a curved surface in computer graphics each face gets "triangulated" down to one or more tris (triangles).  So even though a face may only have four edges, it is really composed of two tris, separated by a hidden edge.  This can cause unpredictability when the face is not planar.

To put an editable edge there you can select the two vertices and hit J to join the two vertices.

Alternatively, you can make the face planar by going to Mesh menu > Clean up > Make Planar Faces.